A New Littlewood-Type Theorem for Bounded Holomorphic Functions in the Unit Disc

Abstract

We prove a new theorem of Littlewood type for bounded holomorphic functions in the unit disc, i.e., we show that these functions do not all admit almost everywhere boundary values along certain systems of tangential approach regions. The novelty of our theorem, in comparison with the previously known results of this kind, is that it also applies to tangential approach regions that are sequential. Indeed, while in the previous results of this kind, the tangential approach regions were required to be curvilinear (Littlewood 1927), (Aikawa 1990), or at least to possess a certain topological property that excludes the possibility that they may be sequential (Di Biase Stokolos Svensson Weiss 2006), our result is the first of this type that also applies to tangential approach regions that are sequential. In order to achieve our result, we have identified a new class of approach regions, called projectively adjacent, that is not defined in topological terms and not depend on their continuous or discrete nature. Indeed, this class is so broad that it contains sequential approach regions and curvilinear ones, as well. Our result can be better appreciated if we recall that Nagel and Stein (1984), elaborating results of Rudin (1979) and Nagel, Rudin and Shapiro (1982), proved the existence of translation invariant systems of tangential and sequential approach regions in the higher-dimensional Euclidean half-spaces along which all bounded holomorphic functions in the unit disc converge a.e. to their nontangential boundary values.